Math
Rørdam K-theory
2.2.10 Liftings
Chapter 2 Projections and Unitary elements
Chapter 3 The K_0 Group of a Unital C -algebra.
Chapter 4 The functor K0
Chapter 5 The ordered abelian group K0
Chapter 6 Inductive limits
Characterization for stable equivalence
Corollary 2.1.4
Definition 2.1.2.
Definition 2.2.1
Definition 2.3.1 (The semigroup of projections of A)
Definition 2.3.3. (The semigroup of projections)
Definition 3.1.4 (The K_0 group of a Unital C-algebra)
Definition 3.1.6 (Stable equivalence)
Definition 3.2.5 (Homotopy equivalence)
Definition 4.1.1 (K0, non-unital case)
Definition 4.2.1 (The Scalar mapping)
Definition 5.1.1.
Definition 5.1.3
Definition 5.1.4 (Positive cone of K0)
Definition 5.16
Definition 6.4.a (Stabilization)
Drawing 2024-02-05 22.07.46
DRAWING
Example 3.1.3
Example 3.3.2
Example 3.3.3
Example 3.3.5
Example 4.1.5
Example 4.3.5
Lemma 2.1.3
Lemma 2.1.5 (Whitehead)
Lemma 2.1.7.
Lemma 2.2.3
Lemma 3.2.7
Lemma 3.2.8
Lemma 4.3.2
Lemma 5.1.2
Lemma 6.3.1
Murray-von Neumann equivalence
Paragraph 3.3.1 Traces and K0
Properties of the Grothendieck group
Proposition 2.1.6.
Proposition 2.1.8
Proposition 2.1.11
Proposition 2.2.5
Proposition 2.2.6
Proposition 2.2.7
Proposition 2.2.8
Proposition 2.3.2.
Proposition 3.1.7 (The standard picture for K0 — the unital case)
Proposition 3.1.8 (Universal property of K0)
Proposition 3.2.4 (Functoriality of K0 for unital C-star Algebras)
Proposition 3.2.6 (Homotopy invariance of K0)
Proposition 4.1.3 (Functoriality of K0)
Proposition 4.1.4 (Homotopy invariance of K0)
Proposition 4.3.2 (Half exactness of K0)
Proposition 4.3.3 (Split exactness of K0)
Proposition 4.3.4 (Direct sums)
Proposition 4.3.8 (Stability)
Proposition 5.1.5
Proposition 5.1.7
Proposition 5.1.9.
Proposition 6.2.4
Proposition 6.4.1 (Stability of K0)
retract
Rørdam K-theory
The functor K00 3.2.3
The functor K0 for unital C-star Algebras 3.2.2
The Grothendieck construction 3.1.1
The group K_00 3.1.5
Theorem 2.1.9. (The polar decomposition)
Theorem 2.1.10 (Carl Neumann Series)
Theorem 4.1.2. (The functor K0)
Theorem 5.2.1
Theorem 5.2.2
Theorem 6.3.2 (Continuity of K0)
Diffuse measure
dirac measures
finite C-star algebra
finite projection
homotopy
infinite C-star algebra
infinite projection
probability measure
quasi-trace
stably finite

Chapter 6 Inductive limits

Proposition 6.2.4
Every inductive sequence of -algebras has an inductive limit , in addition, the following all hold.

  2. (Where the index is {to,from})
  4. If and are as in the uniqueness property of inductive limit, then;
    (a) for all .
    (b) is injective if and only if
    (c) is surjective if and only if

There is a similar result for inductive limits of abelian groups, this can be applied to the compatibility of the functor with inductive limits.

Proposition 6.2.5.
Each sequence of abelian groups has an inductive limit moreover, the following hold.

  2. for each .
  3. Same as in Proposition 6.2.4. No closure needed though.

Lemma 6.3.1
Let be a -algebra,

  1. If is self-adjoint, such that , then there exists a projection such that .
  2. Let . If there exists an element with and then . In the murray von Neumann sense.

Theorem 6.3.2 (Continuity of K0)
For each inductive sequence of -algebras, and are isomorphic as abelian groups.

Definition 6.4.a (Stabilization)
The stabilization of -algebras. We define the stabilization of to be , where denotes the compact operators of for .
Instead of working with tensor products, it is more useful and insightful to work with a different realization of where we build the sequence Where the connecting maps are are given by for each . Let be the limit of its sequence.

Proposition 6.4.1 (Stability of K0)
Let be the canonical inclusion of a -algebra into its stabilization . Then is an isomorphism.

Table Of Contents
Chapter 6 Inductive limits